We study higher order KdV equations from the GL(2,$mathbb{R}$) $cong$ SO(2,1) Lie group point of view. We find elliptic solutions of higher order KdV equations up to the ninth order. We argue that the main structure of the trigonometric/hyperbolic/el
liptic $N$-soliton solutions for higher order KdV equations is the same as that of the original KdV equation. Pointing out that the difference is only the time dependence, we find $N$-soliton solutions of higher order KdV equations can be constructed from those of the original KdV equation by properly replacing the time-dependence. We discuss that there always exist elliptic solutions for all higher order KdV equations.
We study to unify soliton systems, KdV/mKdV/sinh-Gordon, through SO(2,1) $cong$ GL(2,$mathbb R$) $cong$ M{o}bius group point of view, which might be a keystone to exactly solve some special non-linear differential equations. If we construct the $N$-s
oliton solutions through the KdV type B{a}cklund transformation, we can transform different KdV/mKdV/sinh-Gordon equations and the B{a}cklund transformations of the standard form into the same common Hirota form and the same common B{a}cklund transformation except the equation which has the time-derivative term. The difference is only the time-dependence and the main structure of the $N$-soliton solutions has same common form for KdV/mKdV/sinh-Gordon systems. Then the $N$-soliton solutions for the sinh-Gordon equation is obtained just by the replacement from KdV/mKdV $N$-soliton solutions. We also give general addition formulae coming from the KdV type B{a}cklund transformation which plays not only an important role to construct the trigonometric/hyperbolic $N$-soliton solutions but also an essential role to construct the elliptic $N$-soliton solutions. In contrast to the KdV type B{a}cklund transformation, the well-known mKdV/sinh-Gordon type B{a}cklund transformation gives the non-cyclic symmetric $N$-soliton solutions. We give an explicit non-cyclic symmetric 3-soliton solution for KdV/mKdV/sinh-Gordon equations.
We study the Riemann geometric approach to be aimed at unifying soliton systems. The general two-dimensional Einstein equation with constant scalar curvature becomes an integrable differential equation. We show that such Einstein equation includes KdV/mKdV/sine-Gordon equations.
Regarding $N$-soliton solutions, the trigonometric type, the hyperbolic type, and the exponential type solutions are well studied. While for the elliptic type solution, we know only the one-soliton solution so far. Using the commutative B{a}cklund tr
ansformation, we have succeeded in constructing the KdV static elliptic $N$-soliton solution, which means that we have constructed infinitely many solutions for the $wp$-function type differential equation.
We study group theoretical structures of the mKdV equation. The Schwarzian type mKdV equation has the global M{o}bius group symmetry. The Miura transformation makes a connection between the mKdV equation and the KdV equation. We find the special loca
l M{o}bius transformation on the mKdV one-soliton solution which can be regarded as the commutative KdV B{a}cklund transformation can generate the mKdV cyclic symmetric $N$-soliton solution. In this algebraic construction to obtain multi-soliton solutions, we could observe the addition formula.
Recently a new quantum generalization of the Renyi divergence and the corresponding conditional Renyi entropies was proposed. Here we report on a surprising relation between conditional Renyi entropies based on this new generalization and conditional
Renyi entropies based on the quantum relative Renyi entropy that was used in previous literature. Our result generalizes the well-known duality relation H(A|B) + H(A|C) = 0 of the conditional von Neumann entropy for tripartite pure states to Renyi entropies of two different kinds. As a direct application, we prove a collection of inequalities that relate different conditional Renyi entropies and derive a new entropic uncertainty relation.
